xenv: Fit the envelope model in predictor space

In therimalaya/envelope: Computing Envelope Estimators in Multivariate Linear Regression

Description

Fit the envelope model in predictor space with dimension u under linear regression.

Usage

xenv(X, Y, u, asy = TRUE)

Arguments

X	Predictors. An n by p matrix, p is the number of predictors and n is number of observations. The predictors must be continuous variables.
Y	Responses. An n by r matrix, r is the number of responses. The response can be univariate or multivariate and must be continuous variable.
u	Dimension of the envelope. An integer between 0 and p.
asy	Flag for computing the asymptotic variance of the envelope estimator. The default is TRUE. When p and r are large, computing the asymptotic variance can take much time and memory. If only the envelope estimators are needed, the flag can be set to asy = FALSE.

Details

This function fits the envelope model in the predictor space,

Y = μ + ηΩ^{-1}Γ' X +\varepsilon, Σ_{X}=ΓΩΓ'+Γ_{0}Ω_{0}Γ'_{0}

using the maximum likelihood estimation. When the dimension of the envelope is between 1 and p-1, the starting value and blockwise coordinate descent algorithm in Cook et al. (2015) is implemented. When the dimension is p, then the envelope model degenerates to the standard multivariate linear regression. When the dimension is 0, it means that X and Y are uncorrelated, and the fitting is different.

Value

The output is a list that contains the following components:

beta	The envelope estimator of the regression coefficients.
SigmaX	The envelope estimator of the covariance matrix of X.
Gamma	An orthogonal basis of the envelope subspace.
Gamma0	An orthogonal basis of the complement of the envelope subspace.
eta	The estimated eta. According to the envelope parameterization, beta = Gamma * Omega^-1 * eta.
Omega	The coordinates of SigmaX with respect to Gamma.
Omega0	The coordinates of SigmaX with respect to Gamma0.
mu	The estimated intercept.
SigmaYcX	The estimated conditional covariance matrix of Y given X.
loglik	The maximized log likelihood function.
covMatrix	The asymptotic covariance of vec(beta). The covariance matrix returned are asymptotic. For the actual standard errors, multiply by 1 / n.
asySE	The asymptotic standard error for elements in beta under the envelope model. The standard errors returned are asymptotic, for actual standard errors, multiply by 1 / sqrt(n).
ratio	The asymptotic standard error ratio of the standard multivariate linear regression estimator over the envelope estimator, for each element in beta.
n	The number of observations in the data.

References

Cook, R. D., Helland, I. S. and Su, Z. (2013). Envelopes and Partial Least Squares Re- gression. Journal of the Royal Statistical Society: Series B 75, 851 - 877.

Examples

data(wheatprotein)
X <- wheatprotein[, 1:6]
Y <- wheatprotein[, 7]
u <- u.xenv(X, Y)
u

m <- xenv(X, Y, 4)
m
m$beta

therimalaya/envelope documentation built on May 29, 2019, 1:38 p.m.